Model Theory and Differential Equations

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This project centers on algebraic differential equations and model theory. Differential equations are at the heart of modern mathematics and its applications in other sciences. Model theory is a part of mathematical logic in which objects known as definable sets are studied. The notion of a definable set is flexible, changing as one works in different mathematical settings. In the area of differential algebra, definable sets are closely linked with differential equations, providing a connection between the two subjects. This project plans to exploit that link to advance the knowledge of both fields. The project also seeks applications in other areas of mathematics, primarily number theory.This research project addresses problems in the model theory of fields with operators. First, the investigator will generalize his work on the differential algebra of the j-function to understand the differential equations satisfied by analytic covering maps associated with various Shimura varieties. This line of work is expected to have number theoretic consequences related to special-point conjectures as it did in the case of modular curves. The project will also investigate several finiteness results in differential algebra; this line of work is expected to have consequences on effective bounds in number theory and in computational differential algebra. The project aims to develop various aspects of the model theory of supersimple and superstable groups as a generalization of recent work in differential algebraic and difference algebraic groups. The project will investigate the classification of disintegrated strongly minimal sets in differentially closed fields. This work seeks to answer a fundamental question about differential equations: what are the possible structures given by the differential algebraic relations between solutions to a fixed differential equation? Finally, this project aims to adapt and generalize results from algebraic foliations and vector fields for application in differential algebraic geometry.